If people like me -- who are in favour of strong borders and deporting undocumented with criminal convictions -- if these people are disturbed by stories like those above, you're losing the base.
You need to sort out the mess and sort it out pronto! Those with Green Cards, who have no criminal conviction must be released immediately.
You need to have a much better communication flow. Tell the country what you're doing, and why and when you fix mistakes, make sure everyone knows there was a mistake and that you've fixed it.